Invariant generalized functions on $sl(2,R)$ with values in a $sl(2,R)$-module

Abstract

Let $g$ be a finite dimensional real Lie algebra. Let $r:g\to End(V)$ be a representation of $g$ in a finite dimensional real vector space. Let $C_{V}=(End(V)\tens S(g))^{g}$ be the algebra of $End(V)$-valued invariant differential operators with constant coefficients on $g$. Let $U$ be an open subset of $g$. We consider the problem of determining the space of generalized functions $\phi$ on $U$ with values in $V$ which are locally invariant and such that $C_{V}\phi$ is finite dimensional. In this article we consider the case $g=sl(2,R)$. Let $N$ be the nilpotent cone of $sl(2,R)$. We prove that when $U$ is $SL(2,R)$-invariant, then $\phi$ is determined by its restriction to $U\setminus N$ where $\phi$ is analytic. In general this is false when $U$ is not $SL(2,R)$-invariant and $V$ is not trivial. Moreover, when $V$ is not trivial, $\phi$ is not always locally $L^{1}$. Thus, this case is different and more complicated than the situation considered by Harish-Chandra where $g$ is reductive and $V$ is trivial. To solve this problem we find all the locally invariant generalized functions supported in the nilpotent cone $N$. We do this locally in a neighborhood of a nilpotent element $Z$ of $g$ and on an $SL(2,R)$-invariant open subset $U\subset sl(2,R)$. Finally, we also give an application of our main theorem to the Superpfaffian.